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Power-series coefficients of spheroidal eigenvalues
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Numbers
$m$
$n$
$k$ 
$a_{0,n,k}$
0
0
0:
0
0
0
1:
1/3
0
0
2:
-2/135
0
0
3:
4/8505
0
0
4:
26/1913625
0
0
5:
-92/37889775
0
0
6:
513988/9050920003125
0
0
7:
122264/11636897146875
0
0
8:
-57430742/62315584221515625
0
0
9:
-26237052532/1566426840576238265625
0
0
10:
1550889714543116/213229853673440433908203125
0
0
11:
-1220211641925896/4012598155491106347181640625
0
0
12:
-2919216772579320537076/82240105180914082861002635888671875
0
1
0:
2
0
1
1:
3/5
0
1
2:
-6/875
0
1
3:
-4/65625
0
1
4:
458/176859375
0
1
5:
25084/287396484375
0
1
6:
-44433868/28516916162109375
0
1
7:
-17980904/133183399658203125
0
1
8:
693892465438/34132378177388946533203125
0
1
9:
2798408610746228/13823613161842523345947265625
0
1
10:
157872428741905827476/47015836315071652214984893798828125
0
1
11:
-29328649663082524808/106854173443344664124965667724609375
0
1
12:
-146908521610980878030371988/14112067734048028690371425163745880126953125
0
2
0:
6
0
2
1:
11/21
0
2
2:
94/9261
0
2
3:
-21388/44925111
0
2
4:
-3633830/257555661363
0
2
5:
39611204/16226006665869
0
2
6:
-277773545116/4906403669618802351
0
2
7:
-3023903375838584/287775294434151614293203
0
2
8:
1796916847495437790/1949782265352989605605284217
0
2
9:
69602264414158650159764/4156217346277850877033158445654357
0
2
10:
-31661591111502258244649836708/4353118143057764062332604327017232162875
0
2
11:
53071011792043493738557317416/174520463735315813771698046201327216711625
0
2
12:
147150018702964430268686622437033394172/4145505581207061619791854202801212887874720334129525
0
3
0:
12
0
3
1:
23/45
0
3
2:
3302/1002375
0
3
3:
1572668/26387521875
0
3
4:
-3432230674/1293120509484375
0
3
5:
-19759531986908/222578367694998046875
0
3
6:
92707533876516450668/58373052150761068151513671875
0
3
7:
16002955351743561677752/118205430605291163006815185546875
0
3
8:
-15507821133996368111443382/666155569583413745962132489288330078125
0
3
9:
-27689889495056455633364202820844/136785053880410261527134474688019256591796875
0
3
10:
-162624995328055678968065771894129943308/48430270060083672657841450134324122010555267333984375
0
3
11:
70959581208838318150694421093375063278936/258551600843492152439249014467107642279078006744384765625
0
3
12:
13720275821989898533795748997578299805556326588538916/1317971182415675102098148043633276769638817778032415592670440673828125
0
4
0:
20
0
4
1:
39/77
0
4
2:
52674/29674645
0
4
3:
935076/175940970205
0
4
4:
37764832611642/74924536936711587125
0
4
5:
-3187867616210148/241152114584499914320525
0
4
6:
-16139900980217820949844/90612997487168811993405138946875
0
4
7:
10543450108582984881561256/7863305308939022335975704552670865625
0
4
8:
147507800312946862112916928830938/11385212486235892466923138349659698239182078125
0
4
9:
731073112252082046326741972558682452/237727691795752222667276968229813931289954514671875
0
4
10:
-75113032623092883611722703240415182278962147908/7169534805201295305506853746683732579983547474252885431626953125
0
4
11:
-15366093445816524586849273159819741908074557038923944/14495286604273121634425396329653957009152356464422266983923625931640625
0
4
12:
-1421527768251515859818067899122585992919145201174086738611508/134920295145965270943828956805688055339675251884176446501257667627302529833984375
0
5
0:
30
0
5
1:
59/117
0
5
2:
63338/56056455
0
5
3:
15200884/13045065812415
0
5
4:
239831300295254/4156307659965616462125
0
5
5:
16840113131636852/11379139111453864750005825
0
5
6:
-64159557751827254601531908/2113935087634935819874055595593240625
0
5
7:
-1571588854748844615886437784/4133951059233519491179421006867981559375
0
5
8:
8144430040110073537250918102230622/2695038838869340522029969240210425134498473140625
0
5
9:
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0
5
10:
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0
5
11:
109523658797501686573851417846138585238305045776056/4714840886274565500951469518142665595465039782474341106923772966796875
0
5
12:
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0
6
0:
42
0
6
1:
83/165
0
6
2:
60134/76366125
0
6
3:
101060636/276516011165625
0
6
4:
6511710945038/435126977750283328125
0
6
5:
128579438103903484/1362328176463918314943359375
0
6
6:
257272566678133679835777116/63201807304120590732967043538720703125
0
6
7:
-1317175585961621498375552008/22346353296814066009156204679761962890625
0
6
8:
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0
6
9:
836135353079923434391259505956798484626684/319707293800221110121520140751689923260768647213592529296875
0
6
10:
75254962726984787502855728161938489084256407679292/6653404872900072031404445421883020961886775427878261223049163818359375
0
6
11:
-581499195540883891908036792566777245206724261921816/5519469319423127622174089453513622444767429560287651744522571563720703125
0
6
12:
-2076711498511146872017444718843628934380655571796511704428852/353815834169257993878569187459059270893969186951113846611780624891932404041290283203125
0
7
0:
56
0
7
1:
111/221
0
7
2:
1312746/2255916949
0
7
3:
326867588/2313806033828289
0
7
4:
601214767538391430/113535559602330604449197487
0
7
5:
402998256899883487796/27725951332687145259516272312835
0
7
6:
71231080656748135820034913363612/234771121771711437604277699144322257661609915
0
7
7:
58404393505703940237300542775837544/9500778125574645468482436743237927377344857311341
0
7
8:
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0
7
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0
7
10:
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0
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0
7
12:
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0
8
0:
72
0
8
1:
143/285
0
8
2:
944302/2106570375
0
8
3:
2720719948/43289863213471875
0
8
4:
13055088192218326/5823562138863505896234375
0
8
5:
7675609554029860756/2365094173645941332108185546875
0
8
6:
52367670487519154091900560153284/984325482533212126704419334298435325830078125
0
8
7:
116854700951844411079751972213439848/354072422411652114962651467611443241365284423828125
0
8
8:
1250314176683721008935588887061609744766378/151554892820870085779504584675528843589426058397613525390625
0
8
9:
-392540607856042534581959844947537646815013337853924/5200494718686533841106181729087458355321838722246062154398956298828125
0
8
10:
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0
8
11:
56624108514922979650249368820009707637178485416766577699944/35928102959861279588410529157228963969079278747220140310433957774215221405029296875
0
8
12:
73981719632316681085034655049654445317093621596787668032582698216928556052/15856913139811817264728906584332265797664618739402191156884279367214797750416251759922504425048828125
0
9
0:
90
0
9
1:
179/357
0
9
2:
1864018/5232418695
0
9
3:
268520116/8669264893367715
0
9
4:
172137295132137094/160733747844411717216866625
0
9
5:
15462968876632443164/16973580212618583985131445719975
0
9
6:
3302403491776742376207340071668/249805915304575711111120341626562225247584375
0
9
7:
9248734395863553959057802580949176/222862598697570088637808234939746103319055667065625
0
9
8:
4436578580444715592537926553386549994986238/9090435071940478570036212956125491965561790192448022107828125
0
9
9:
161182161453637074796063864764558030847178210584028/21733555717055517177339677377343616368908756178333660202112585963921875
0
9
10:
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0
9
11:
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0
9
12:
8984494941774119339883343866034985096124279589098088883777030848059942052436212/9633813987682339338888567496807217592331224481687999809345647562768770277029449237487914686983851904296875
0
10
0:
110
0
10
1:
219/437
0
10
2:
10288734/35467717525
0
10
3:
1006347964/60959110932285525
0
10
4:
443085063003377637158/792721630338748533355068703125
0
10
5:
3101359339027359667673444/10324474529047743963040576654394728125
0
10
6:
5263680199082897077035068834723113916/1298094243116896549757662445469226441151555244140625
0
10
7:
1859079122367026381895978734405437285976/247895759513790617210671039548812704240271353418291015625
0
10
8:
606048623617748320632648900251593605516829328596000606/8993407082607138632787833374106544777955313904059565287272582847900390625
0
10
9:
446131653119192818429934535082358360307146454155669122906468/1252029766768475537364782758930799754532282940548431968918620090059022789306640625
0
10
10:
6056707143569492514689074770946259037687221322705813255262421448632257375204/1029480221576399270739185503802080014975291839817035139149766002889964970236223932961688995361328125
0
10
11:
-1512710847877991442834685028077365194874022598709379685653677572830373514092776/40448558673069884547143709131339306173837846011458533444959528917002057307391302750773750179290771484375
0
10
12:
-225782632603123581790611528851517363416855976353929547286653314648368012461578536304921376796484/1590181507580679027725660016984448798232456333017745163281867213150497760897517891131762894267991061488909237384796142578125
0
11
0:
132
0
11
1:
263/525
0
11
2:
662278/2749359375
0
11
3:
3497866492/373591543623046875
0
11
4:
71978531359405714/230470070928289156494140625
0
11
5:
725609490150072129596/6488452715602990658924102783203125
0
11
6:
75325691347208729622483878740156/52304558022628959549100638878152370452880859375
0
11
7:
489471351881144188411110701061169832/287219918883973900516284897693446423113346099853515625
0
11
8:
4735696595267172756116512664760491940142047658/357031968087468986874131717791997305954878238029778003692626953125
0
11
9:
23057190298516887154007723335535689774134130483800243956/588375602961821057800097833435266208980340057132146792369894683361053466796875
0
11
10:
77295151080518723739325668679189073839793803709504296841096434178396/257389602949698349307052958218875729597714559733914735457175124694549595005810260772705078125
0
11
11:
22451634003512987427796196315596457530903307502673298491206944502163112/6449364483000964320705133782643420270033643229696386269125808522175930193043313920497894287109375
0
11
12:
-212687055709390566773778827320615695799318083906131102381102307254820584079671287855708/11070123338172096652636187098952356952121704499502861375028589673161152185135033696070649966713972389698028564453125
Definition
The coefficients $a_{0,n,k}$ are defined by $\chi_n(\sqrt q)=\sum_{k\geq0}a_{0,n,k}q^k$ near $q=0$. Here $\chi_n(\sqrt q)$ is the $n$th eigenvalue of $-((1-x^2)\psi')'+qx^2\psi=\chi\psi$ with $\psi$ bounded at $x=\pm1$, indexed increasingly from $n=0$ [1].
Parameters
$m$
—   angular order ($m=0$)
$n$
—   mode index ($n\geq0$)
$k$
—   power of $q=c^2$ ($k\geq0$)
Formulas
(1)
$\chi_n(\sqrt q)=\lambda_n^0(q)+q$. If $\lambda_n^0(q)=\sum_{k\geq0}\ell_{2k}q^k$ in the DLMF notation, then $a_{0,n,1}=\ell_2+1$ and $a_{0,n,k}=\ell_{2k}$ for $k\ne1$ [2] [1].
(2)
$a_{0,n,0}=n(n+1)$ and $a_{0,n,1}=\frac12\left(1+\frac{1}{(2n-1)(2n+3)}\right)$ [1].
(3)
$a_{0,n,2}=\frac{n^2(n-1)^2}{2(2n-3)(2n-1)^3(2n+1)}-\frac{(n+1)^2(n+2)^2}{2(2n+1)(2n+3)^3(2n+5)}$ [1].
(4)
$\chi_0(\sqrt q)=\frac{q}{3}-\frac{2q^2}{135}+\frac{4q^3}{8505}+\frac{26q^4}{1913625}+O(q^5)$ as $q\to0$.
Comments
(5)
The expansion variable is $q=c^2$ with $x\in[-1,1]$. These are Taylor coefficients, so the coefficient of $q^k$ is the $k$th derivative with respect to $q$ divided by $k!$; no odd powers of $c$ occur. The angular order is fixed at $m=0$ and is distinct from the mode index $n$. The analytic branch starts at $\chi_n(0)=n(n+1)$.
(6)
At $q=0$ the bounded eigenfunctions are scalar multiples of the Legendre polynomials $P_n(x)$. Their normalization does not affect the eigenvalue coefficients.
(7)
The series describes the local dependence on bandwidth of the prolate spheroidal differential eigenvalues. Small-parameter eigenvalue computation is discussed in [3]. The coefficient list alone does not supply a remainder bound for an evaluation at nonzero $q$.
Programs
(P1)
Sage
# Download the attached generate.py into the working directory.
from generate import coefficient
coefficient(0, 3)  # a_(0,0,3) = 4/8505
Links
Similar tables
Prolate spheroidal differential eigenvalues —   These coefficients expand the differential eigenvalues at zero bandwidth.
Slepian concentration eigenvalues —   The concentration operator shares the order-zero spheroidal eigenfunctions with the differential operator; its eigenvalues are a different quantity.
Legendre polynomials —   These are the bounded eigenfunctions at the expansion point.
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (it holds every coefficient with $m=0$, $0\leq n\leq11$ and $0\leq k\leq12$, covering the first twelve spectral modes and their first twelve corrections at the Legendre limit $q=0$ in the normalization $q=c^2$; these finite initial segments keep the exact fractions compact)
How they were obtained:

All arithmetic is over the rational field. The generator uses perturbation of the operator diagonal in Legendre polynomials by multiplication by $x^2$. At Taylor order $r$, the eigenfunction correction has degree at most $n+2r$, so its finite support is exact rather than a numerical truncation. The coefficient of $P_n$ is fixed at one as a computational normalization. The attached argument-selection.md gives the recurrence and its finite-support argument.

more

The attached checks independently solve the differential equation in a monomial basis with a different eigenvector normalization. They compare all 156 coefficients, test the explicit formulas through $k=4$ in DLMF 30.3.9 through 30.3.12 for every selected mode, and verify the differential equation coefficient by coefficient through $q^{12}$. Known values for the first two modes and the first three Legendre polynomials control the checks. The same checks can be applied to the coefficients read from the database. Exact rational values need no decimal precision or error estimate.



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